Parallel
Lines
| The Poincaré disc model of Hyperbolic
2 space is the interior of the blue circle shown. The points on the circle
can be thought of as the horizon of the space or points at infinity. The
red dot is simply a reference point to us as observers of the space. It
makes no more sense to speak of the centre of the Hyperbolic plane than
it does to talk of a particular point being the centre of the Euclidean
plane
A line through two points is the unique arc of the circle through the points that cuts the horizon at right angles as shown. If we move the line by moving the green point marked, keeping the point inside the disc, we see that, in particular, a d-line (d standing for disc) that passes through our centre point is a segment of a Euclidean straight line. |
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